Numerical solutions of fuzzy differential equations by extended Runge–Kutta-like formulae of order 4

Dynamical control of accuracy in the fuzzy Runge-Kutta methods to estimate the solution of a fuzzy differential equation

“…For Runge-Kutta of order m, let [20], [24] y(t, r))], i = 1, 2, ..., m (2.14) Equation (2.13) is to be exact for powers of h through h m , m ≥ 1. Therefore, the truncation error T m can be written as…”

“…It is proved, the common significant digits of two sequential results is almost equal to the common significant digits between the approximate and the exact solution. In this theorem, the notation C a,b means the common significant digits between two distinct real numbers a and b in R which is defined as [1]- [5], [17] 20) and by step size h 2 we have…”

“…Consider the fuzzy initial value problem [6], [20]. By using the Runge-Kutta method of order 4, we have y 1 (t n+1 ; r) = y 1 (t n ; r)…”

“…One of the fundamental difference schemes to solve FDEs is Runge-Kutta methods which play the important role in numerical methods of solving ordinary differential equations [8], [16], [19], [21], [23], [36], [38]. In recent years, some works were done to solve FDEs, for example, Abbasbandy and Allahviranloo [6], Ghazanfari and Shakerami [20], Jayakumar etal. [24], Palligkinis etal.…”

Fourth Order Diagonally Implicit Multistep Block Method for Solving Fuzzy Differential Equations

“…Meanwhile, Abbasbandy and Viranloo [10] apply the fourth order RK method to solve FDEs while Palligkinis et al [11] propose RK method for a generalized problem and show the convergence of the method. Ghazanfari and Shakerami [12] present the extended RK-like formulae to increase the accuracy of the solutions. Meanwhile, Allahviranloo et al (see [13], [14], [15]) use predictor-corrector method for solving FDEs by considering the Adams methods.…”